Scott Doyle /<EMAIL_ADDRESS>The posterior probability is:
$ P(\omega_{i} | \mathbf{x}) = \frac{p(\mathbf{x}|\omega_{i})P(\omega_{i})}{\sum_{j=1}^{c}p(\mathbf{x}|\omega_{j}) P(\omega_{j})}$
$P(\omega_{i})$: The prior probability of $\omega_{i}$
$p(\mathbf{x}|\omega_{i})$: The class-conditional PDF
If we know these two quantities, then we have our decision rule:
If $p(\mathbf{x}|\omega_{1}) P(\omega_{1}) > p(\mathbf{x}|\omega_{2})P(\omega_{2})$, decide $\omega_{1}$ otherwise, decide $\omega_{2}$
We are rarely given $P(\omega_{i})$ and $p(\mathbf{x}|\omega_{i})$ directly.
Instead, we collect a limited, random sampling of $\mathbf{x}$, which constitutes our training data.
Training data is how we estimate $P(\omega_{i})$ and $p(\mathbf{x}|\omega_{i})$.
We can estimate $P(\omega_{i})$ simply as the proportion of our training data with the class label $\omega_{i}$.
Estimating $p(\mathbf{x}|\omega_{i})$ is much more difficult:
If we can make assumptions about the form of $p(\mathbf{x}|\omega_{i})$, things are much easier.
The normal distribution is written as:
$ p(x) \sim N(\mu,\sigma^{2}) = \frac{1}{\sqrt{2}\sigma}\exp{\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]}$
Its parameters, the mean and variance, are defined as:
\begin{align} \mu\equiv \mathcal{E}\left[x\right] &= \int_{-\infty}^{\infty}(x)p(x)dx \\ \sigma^{2}\equiv \mathcal{E}\left[(x-\mu)^{2}\right]&=\int_{-\infty}^{\infty}(x-\mu)^{2}p(x)dx\\ \end{align}
We reduce the problem from estimating:
In general, we write a set of parameters as $\boldsymbol{\theta}_{i}$, where $i\in\{1,\ldots,c\}$ refers to the class.
If $p(\mathbf{x}|\omega_{i}) \sim N(\boldsymbol{\mu}_{i},\boldsymbol{\Sigma}_{i})$, then $\boldsymbol{\theta}_{i}=(\boldsymbol{\mu}_{i}, \boldsymbol{\Sigma}_{i})$.
Note that “parameters” can refer to any set of parameters for a statistical model, including the values governing how classifiers behave.
How do we find the parameters of these distributions?
Maximum Likelihood Estimation (MLE)
Bayesian Estimation (BAY)
Suppose we have $c$ sets of training data for each class: $\mathcal{D}_{1},\ldots,\mathcal{D}_{c}$
Each sample in $\mathcal{D}_{i}$ is drawn independently according to probability law $p(\mathbf{x}|\omega_{i})$.
These samples are independent and identically distributed (i.i.d.) random variables.
We’re assuming that $p(\mathbf{x}|\omega_{i})$ has some known parametric form (e.g. it’s a Gaussian), which means it’s defined by the parameter set $\boldsymbol{\theta}_{i}$.
Therefore we write $p(\mathbf{x}|\omega_{i})$ as $p(\mathbf{x}|\omega_{i}, \boldsymbol{\theta}_{i})$, where we have a different $\boldsymbol{\theta}$ (parameter set) for each class.
We can treat each class independently by assuming that samples in $\mathcal{D}_{i}$ give no information about $\boldsymbol{\theta}_{j}$ if $i\neq j$.
So we now have $c$ problems, stated thusly:
Use a set of training data $\mathcal{D}$ drawn independently from the probability density $p(\mathbf{x}|\boldsymbol{\theta})$ to estimate the unknown parameter vector $\boldsymbol{\theta}$.
Once we find the parameter values, we use those to model our distribution, and then we can use that to create our PDFs.
Suppose $\mathcal{D} = \{\mathbf{x}_{1}, \ldots, \mathbf{x}_{n}\}$.
Since these samples are i.i.d., we can write:
$ p(\mathcal{D}|\boldsymbol{\theta}) = \prod_{k=1}^{n}p(\mathbf{x}_{k}|\boldsymbol{\theta})$
This is the likelihood of observing a particular $\mathcal{D}$ given parameter values $\boldsymbol{\theta}$.
This is equivalent to the product of the likelihoods of observing each individual sample $\mathbf{x}_{k}$ in the training set, given the parameter values $\boldsymbol{\theta}$.
This is the combined likelihood:
$p(\mathcal{D}|\boldsymbol{\theta}) = \prod_{k=1}^{n} p(\mathbf{x}_{k}|\boldsymbol{\theta})$
This is a function of $\boldsymbol{\theta}$, not $x$.
The maximum-likelihood estimate of $\boldsymbol{\theta}$ is the value $\widehat{\theta}$ that maximizes the likelihood of producing the observed $\mathcal{D}$.
Whenever we want to maximize something, we set the gradient to $0$.
If $\boldsymbol{\theta} = (\theta_{1}\ldots,\theta_{p})^{t}$, then the gradient operator is:
$\boldsymbol{\nabla}_{\boldsymbol{\theta}}\equiv \begin{bmatrix} \frac{\partial}{\partial\theta_{1}}\\ \vdots\\ \frac{\partial}{\partial\theta_{p}} \end{bmatrix}$
Let’s make things easier by converting to log-likelihoods:
$ l(\boldsymbol{\theta}) \equiv \ln{p(\mathcal{D}|\boldsymbol{\theta})} = \sum_{k=1}^{n}\ln{p(\mathbf{x}_{k}|\boldsymbol{\theta})} $
We need the value of $\boldsymbol{\theta}$ such that: $\widehat{\boldsymbol{\theta}}=\textrm{arg max}_{\boldsymbol{\theta}}\left[l(\boldsymbol{\theta})\right]$
Set the gradient of $l$ to zero:
$ \boldsymbol{\nabla}_{\boldsymbol{\theta}} l = \sum_{k=1}^{n}\boldsymbol{\nabla}_{\boldsymbol{\theta}}\ln{p(\mathbf{x}_{k}|\boldsymbol{\theta})} = \mathbf{0}$
This is a system of equations, one for each partial derivative.
As with all gradient methods, you need to make sure the solution is a global maximum and not a minimum, local solution, or inflection point.
$\widehat{\boldsymbol{\theta}}$ is only an estimate; to get the true value, we’d have to collect $n=\infty$ training points.
Central Limit Theorem: as $n\rightarrow\infty$, if the underlying distribution has a well-defined expected value and variance, the arithmetic mean of the collected samples will be normally distributed.
If you have a large $n$, it’s a good guess for the parametric distribution.
Let’s examine the Gaussian cases under a few conditions, just like we did with linear discriminants, with the goal of estimating $\boldsymbol{\theta}=(\boldsymbol{\mu}, \boldsymbol{\Sigma})$.
Note: Some of the results in this section may seem obvious; however, this is to illustrate the mathematical justification for what comes naturally. (You’ll see what I mean…)
Let’s say you already know the variance of a random variable. How would you estimate the mean? (What is the average radius of a nucleus?)
Restated: Given a random nucleus, what do we expect its radius to be?
Why does this give us the optimal estimate of $\boldsymbol{\theta}$?
Consider a single sample $\mathbf{x}_{k}$. We are assuming we have the covariance and need to find $\boldsymbol{\mu}$.
If $p(\mathbf{x}_{k}|\boldsymbol{\mu})$ is normally distributed, and we are using the log-likelihood, we can write:
$ l(\boldsymbol{\mu}) = \ln{p(\mathbf{x}_{k}|\boldsymbol{\mu})} = -\frac{1}{2}\ln{\left[(2\pi)^{d}|\boldsymbol{\Sigma}|\right]} - \frac{1}{2}(\mathbf{x}_{k} - \boldsymbol{\mu})^{T}\boldsymbol{\Sigma}^{-1}(\mathbf{x}_{k} - \boldsymbol{\mu}) $
Our gradient function, then, is the derivative of that with respect to $\boldsymbol{\mu}$:
$ \boldsymbol{\nabla}_{\boldsymbol{\mu}}\ln{p(\mathbf{x}_{k}|\boldsymbol{\mu})} = \boldsymbol{\Sigma}^{-1}(\mathbf{x}_{k} - \boldsymbol{\mu}) $
$ \boldsymbol{\nabla}_{\boldsymbol{\mu}} \ln{p(\mathbf{x}_{k}|\boldsymbol{\mu})} = \boldsymbol{\Sigma}^{-1}(\mathbf{x}_{k} - \boldsymbol{\mu}) $
Once again, we find our optimal mean $\widehat{\boldsymbol{\mu}}$ by setting this equal to zero:
\begin{align} \sum_{k=1}^{n}\boldsymbol{\Sigma}^{-1}(\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}}) &= \mathbf{0} \\ \sum_{k=1}^{n}(\mathbf{x}_{k}) - \sum_{k=1}^{n}(\widehat{\boldsymbol{\mu}}) &= \mathbf{0}\\ \sum_{k=1}^{n}(\mathbf{x}_{k}) - n\widehat{\boldsymbol{\mu}} &= \mathbf{0}\\ \widehat{\boldsymbol{\mu}} &= \frac{1}{n}\sum_{k=1}^{n}\mathbf{x}_{k} \end{align}
More often, you won’t know anything about the distribution ahead of time, so you’ll need to estimate both $\boldsymbol{\Sigma}$ and $\boldsymbol{\mu}$.
Let’s start with the univariate case, so $\boldsymbol{\theta}=(\mu,\sigma^{2})$.
Here the log-likelihood of a single point $x_{k}$ is:
$ \ln{p(x_{k}|\boldsymbol{\theta})} = - \frac{1}{2} \ln{\left[2\sigma^{2}\right]} - \frac{1}{2\sigma^{2}} (x_{k} - \mu)^{2} $
And the derivative is:
$ \boldsymbol{\nabla}_{\boldsymbol{\theta}} \ln{ p(x_{k}|\boldsymbol{\theta}) } = \begin{bmatrix} \frac{\partial}{\partial\mu} \ln{p(x_{k}|\boldsymbol{\theta})}\\ \frac{\partial}{\partial\sigma} \ln{p(x_{k}|\boldsymbol{\theta})} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sigma^{2}} (x_{k} - \mu)\\ -\frac{1}{\sigma^{2}} + \frac{(x_{k} - \mu)^{2}}{2\sigma^{2}} \end{bmatrix}$
This is a system of equations, one for each of the parameters we’re interested in. We set both of those to zero to solve for the optimal estimates.
$\begin{bmatrix} \frac{1}{\sigma^{2}} (x_{k} - \mu)\\ -\frac{1}{\sigma^{2}} + \frac{(x_{k} - \mu)^{2}}{2\sigma^{2}} \end{bmatrix}=\mathbf{0}$
\begin{align} \sum_{k=1}^{n} \frac{1}{\widehat{\sigma}^{2}} (x_{k} - \widehat{\mu}) &= 0 \\ - \sum_{k=1}^{n} \frac{1}{\widehat{\sigma}^{2}} + \sum_{k=1}^{n} \frac{(x_{k} - \widehat{\mu})^{2}}{\widehat{\sigma}^{2}} &= 0 \end{align}
We rearrange a bit, and end up with:
\begin{align} \widehat{\mu} &= \frac{1}{n} \sum_{k=1}^{n} x_{k} \text{(just like before)} \\ \widehat{\sigma}^{2} &= \frac{1}{n} \sum_{k=1}^{n} (x_{k} - \widehat{\mu})^{2} \end{align}
Once again, $\widehat{\sigma}$ should look familiar!
Extending to the multivariate case works the same way, with just more manipulations.
Assuming you trust me (or if not, assuming you trust the textbook), the result of these manipulations is:
$ \widehat{\boldsymbol{\mu}} = \frac{1}{n} \sum_{k=1}^{n} \mathbf{x}_{k} $
$ \widehat{\boldsymbol{\Sigma}} = \frac{1}{n} \sum_{k=1}^{n} (\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}}) (\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}})^{T} $
(Reminder: $(\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}}) (\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}})^{T}$ is a matrix.)
What can go wrong with Maximum Likelihood Estimation?
Refer back to our assumptions…
Model: If the model is not correct and the samples are not normally distributed, we have to solve using a different formula and our parameters might not even be correct.
Sample Size: If there are too few training samples for the Central Limit Theorem to come into play, then we may not get an accurate estimate.
This is not just a problem with MLE, but a problem with ALL estimation attempts – so much so that we will look at model selection in another lecture.
The MLE for $\sigma^{2}$ is biased, meaning that it systemically underestimates the actual variance:
$ \mathcal{E} \left[ \frac{1}{2} \sum_{k=1}^{n} (x_{k} - \widehat{\mu})^{2}\right] = \frac{n-1}{n} \sigma^{2} \neq \sigma^{2} $
In other words, our estimate (on the left) is a fraction of the true variance.
This is an asymptotically unbiased estimator: it gets less biased as $n\rightarrow\infty$.
We can use an absolutely unbiased estimator called the sample covariance matrix:
$ \mathbf{C} = \frac{1}{n-1} \sum_{k=1}^{n} (\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}}) (\mathbf{x}_{k} - \widehat{\boldsymbol{\mu}})^{T} $
In MLE, we assumed that $\boldsymbol{\theta}$ was fixed, we just didn’t know what it was.
In BAY, we assume that $\boldsymbol{\theta}$ is a random variable, just like $\mathbf{x}$, and that training data allows us to convert the distribution of $\boldsymbol{\theta}$ into a posterior probability density.
The results of MLE and BAY will very often be identical – so why use Bayes?
$ P(\omega_{i}|\mathbf{x}) = \frac{p(\mathbf{x}|\omega_{i}) P(\omega_{i})}{\sum_{j=1}^{c} p(\mathbf{x}|\omega_{j}) P(\omega_{j})} $
Recall the main components of Bayesian Classification:
$P(\omega_{i}|\mathbf{x})$: Posterior probability, what we want to obtain.
$p(\mathbf{x}|\omega_{i})$: Class-conditional Density, what we get from our training data values.
$P(\omega_{i})$: Prior Probability, what we get from the fraction of each class in our training.
How do we obtain these quantities? We estimate them using as much information as we can about the problem – our prior knowledge, and our training data.
We can rewrite Bayes to show its dependence on the training set $\mathcal{D}$:
$ P(\omega_{i}|\mathbf{x},\mathcal{D}) = \frac{ p(\mathbf{x}|\omega_{i},\mathcal{D}) P(\omega_{i}|\mathcal{D}) }{ \sum_{j=1}^{c}p(\mathbf{x}|\omega_{j},\mathcal{D}) P(\omega_{j}|\mathcal{D}) } $
Assume prior probabilities are known ahead of time OR can be calculated (fraction of $\mathcal{D}$ belonging to each class); thus $P(\omega_{i}) = P(\omega_{i}|\mathcal{D})$.
As we did before, we break up the training set into $c$ class-based subsets, $\mathcal{D}_{1},\ldots,\mathcal{D}_{c}$.
Again, the samples in $\mathcal{D}_{i}$ have no useful information about $p(\mathbf{x}|\omega_{j},\mathcal{D})$ if $i\neq j$.
Therefore we can handle each class as an independent estimation problem:
Use a set of training data $\mathcal{D}$ drawn independently according to the fixed but unknown probability distribution $p(\mathbf{x})$ to determine $p(\mathbf{x}|\mathcal{D})$.
In other words, we’re trying to determine the probability of observing a feature vector given the set of training we’ve already observed.
We’re assuming that even though we don’t know $p(\mathbf{x})$, it has a parametric form. So again, we are estimating the parameter vector $\boldsymbol{\theta}$.
This is expressed by saying that the function $p(\mathbf{x}|\boldsymbol{\theta})$ is “completely known”.
Information about $\boldsymbol{\theta}$ without observing the samples (i.e. prior) is contained in a known density $p(\boldsymbol{\theta})$.
After we observe the training set, this gets converted to a posterior density $p(\boldsymbol{\theta}|\mathcal{D})$, which is hopefully peaked around the true value of $\boldsymbol{\theta}$.
So we’ve connected $p(\mathbf{x})$ (what we want) to $\boldsymbol{\theta}$ (the parameter vector) by $p(\mathbf{x}|\boldsymbol{\theta})$, and we’ve connected the parameter vector to the training set by $p(\boldsymbol{\theta}|\mathcal{D})$.
In this way we can estimate $p(\mathbf{x})$ by calculating $p(\mathbf{x}|\mathcal{D})$, and we do THAT by integrating the joint density $p(\mathbf{x}, \boldsymbol{\theta} | \mathcal{D})$ over the entire parameter space for $\boldsymbol{\theta}$:
\begin{align} p(\mathbf{x} | \mathcal{D}) &= \int p(\mathbf{x}, \boldsymbol{\theta} | \mathcal{D}) d\boldsymbol{\theta} \\ p(\mathbf{x} | \mathcal{D}) &= \int p(\mathbf{x} | \boldsymbol{\theta}, \mathcal{D}) p(\boldsymbol{\theta} | \mathcal{D}) d\boldsymbol{\theta} \end{align}
Since $\mathbf{x}$ and $\mathcal{D}$ are independent, we can drop the $\mathcal{D}$ in the first term, and write:
$ p(\mathbf{x} | \mathcal{D}) = \int p(\mathbf{x} | \boldsymbol{\theta}) p(\boldsymbol{\theta} | \mathcal{D}) d\boldsymbol{\theta} $
$ p(\mathbf{x} | \mathcal{D}) = \int p(\mathbf{x} | \boldsymbol{\theta}) p(\boldsymbol{\theta} | \mathcal{D}) d\boldsymbol{\theta} $
This equation links the class-conditional density $p(\mathbf{x} | \mathcal{D})$ to the posterior density for the parameter vector, $p(\boldsymbol{\theta} | \mathcal{D})$, and to the likelihood $p(\mathbf{x} | \boldsymbol{\theta})$.
If $p(\boldsymbol{\theta} | \mathcal{D})$ peaks sharply around a specific value $\widehat{\boldsymbol{\theta}}$, then the above equation becomes:
$ p(\mathbf{x}|\mathcal{D}) \approx p(\mathbf{x} | \widehat{\boldsymbol{\theta}})$
As the number of samples increases, we will see this “peakiness” increase as well.
The summary of the basic assumptions for BAY estimation are as follows:
We want to compute the posterior density $p(\boldsymbol{\theta} | \mathcal{D})$, because that lets us find $p(\mathbf{x} | \mathcal{D})$.
By Bayes formula we have:
$ p(\boldsymbol{\theta} | \mathcal{D}) = \frac{p(\mathcal{D} | \boldsymbol{\theta}) p(\boldsymbol{\theta})} { \int p(\mathcal{D} | \boldsymbol{\theta}) p(\boldsymbol{\theta}) d\boldsymbol{\theta}} $
Since the samples themselves are independent (we assume), we can also write:
$ p(\mathcal{D} | \boldsymbol{\theta}) = \prod_{k=1}^{n} p(\mathbf{x}_{k} | \boldsymbol{\theta}) $
\begin{align} p(\mathbf{x} | \mathcal{D}) &= \int p(\mathbf{x} | \boldsymbol{\theta}) p(\boldsymbol{\theta} | \mathcal{D}) d\boldsymbol{\theta} \\ p(\boldsymbol{\theta} | \mathcal{D}) &= \frac{p(\mathcal{D} | \boldsymbol{\theta}) p(\boldsymbol{\theta})} { \int p(\mathcal{D} | \boldsymbol{\theta}) p(\boldsymbol{\theta}) d\boldsymbol{\theta}} \end{align}
Suppose that $p(\mathcal{D} | \boldsymbol{\theta})$ reaches a sharp peak when $\boldsymbol{\theta} = \widehat{\boldsymbol{\theta}}$.
Assuming $p(\boldsymbol{\theta})$ is not zero at $\widehat{\boldsymbol{\theta}}$, then $p(\boldsymbol{\theta} | \mathcal{D})$ also peaks around that point.
Thus, $p(\mathbf{x} | \mathcal{D}) \approx p(\mathbf{x} | \widehat{\boldsymbol{\theta}})$, which is what we get using MLE.
However, the Bayesian equations encode much more information about the problem – including uncertainty in the estimates – versus MLE.
This section may be a little confusing. All we’re saying is the following:
Assume that $p(x | \mu) \sim N(\mu, \sigma^{2})$ and we have a known $\sigma^{2}$.
We also know the density $p(\mu)$, which contains any prior knowledge about $\mu$ – specifically, we know that it too is Gaussian: $p(\mu) \sim N(\mu_{0}, \sigma_{0}^{2})$.
In this setup, $\mu_{0}$ is our “best guess” as to the value of $\mu$, and $\sigma_{0}^{2}$ represents the variance or “uncertainty” about our guess.
Now we have to start drawing values of $\mu$ from $p(\mu)$, where each value completely determines the density for $x$. THEN, we take a bunch of samples of $x$, $\mathcal{D} = \{x_{1}, \ldots, x_{n}\}$, which are governed by that density.
\begin{align} p(\mu | \mathcal{D}) &= \frac{ p(\mathcal{D} | \mu) p(\mu) }{ \int p(\mathcal{D} | \mu) p(\mu) d\mu} \\ p(\mathbf{x}) &= \prod_{i=1}^{d}\frac{1}{\sqrt{2}\sigma_{i}}\exp{\left[-\frac{1}{2}\left(\frac{x_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}\right]} \\ p(\mu | \mathcal{D}) &= \alpha \prod_{k=1}^{n} p(x_{k} | \mu) p(\mu) \end{align}
(Where $\alpha$ is a normalization factor that depends just on $\mathcal{D}$.)
So think about this carefully: Our prior guess about $\mu$ is directly influenced by our observed training samples $x_{k}$.
$p(\mu | \mathcal{D}) = \alpha \prod_{k=1}^{n} p(x_{k} | \mu) p(\mu) $
Since we know that both $p(x_{k} | \mu) \sim N(\mu, \sigma^{2})$ and $p(\mu) \sim N(\mu_{0}, \sigma_{0}^{2})$, we can rewrite the above equation as:
$ p(\mu | \mathcal{D}) = \alpha \prod_{k=1}^{n} \frac{ 1 }{ \sqrt{2} \sigma_{~} } \exp{ \left[ -\frac{1}{2} \left( \frac{ x_{k} - \mu_{~} }{ \sigma_{~} } \right)^{2} \right] } \frac{ 1 }{ \sqrt{2} \sigma_{0} } \exp{ \left[ -\frac{1}{2} \left( \frac{ \mu - \mu_{0} }{ \sigma_{0} } \right)^{2} \right] } $
\begin{align} p(\mu | \mathcal{D}) &= \alpha \prod_{k=1}^{n} \frac{ 1 }{ \sqrt{2} \sigma_{~} } \exp{ \left[ -\frac{1}{2} \left( \frac{ x_{k} - \mu_{~} }{ \sigma_{~} } \right)^{2} \right] } \frac{ 1 }{ \sqrt{2} \sigma_{0} } \exp{ \left[ -\frac{1}{2} \left( \frac{ \mu - \mu_{0} }{ \sigma_{0} } \right)^{2} \right] } \\ p(\mu | \mathcal{D}) &= \alpha^{\prime} \exp{ \left[ -\frac{1}{2} \left( \sum_{k=1}^{n} \left( \frac{ \mu - x_{k}}{ \sigma } \right)^{2} + \left( \frac{ \mu - \mu_{0} }{ \sigma_{0} } \right)^{2} \right) \right] } \\ p(\mu | \mathcal{D}) &= \alpha^{\prime\prime} \exp{ \left[ -\frac{1}{2} \left[ \left( \frac{n}{\sigma^{2}} + \frac{1}{\sigma_{0}^{2}} \right) \mu^{2} - 2 \left( \frac{1}{\sigma^{2}} \sum_{k=1}^{n} x_{k} + \frac{\mu_{0}}{\sigma_{0}^{2}} \right) \mu \right] \right] } \end{align}
Factors independent of $\mu$ are collected into the constants $\alpha$, $\alpha^{\prime}$, and $\alpha^{\prime\prime}$.
The takeaway message from this derivation is that $p(\mu | \mathcal{D})$ is an exponential function of a quadratic function of $\mu$. In other words, $p(\mu | \mathcal{D})$ is a normal density.
As we increase our training set size, $p(\mu | \mathcal{D})$ will always be normal.
$p(\mu | \mathcal{D})$ is called a reproducing density, meaning it reproduces the same form as the sampling is increased.
$p(\mu)$ is a conjugate prior, meaning that it has the same form as the associated prior.
If we write the parameters as $p(\mu | \mathcal{D}) \sim N(\mu_{n}, \sigma_{n}^{2})$, then we can simplify.
$ p(\mu | \mathcal{D}) = \alpha^{\prime\prime} \exp{ \left[ -\frac{1}{2} \left[ \left( \frac{n}{\sigma^{2}} + \frac{1}{\sigma_{0}^{2}} \right) \mu^{2} - 2 \left( \frac{1}{\sigma^{2}} \sum_{k=1}^{n} x_{k} + \frac{\mu_{0}}{\sigma_{0}^{2}} \right) \mu \right] \right] } $
Let:
\begin{align} \frac{ 1 }{ \sigma_{n}^{2} } &= \frac{ n }{ \sigma^{2} } + \frac{ 1 }{ \sigma_{0}^{2} }\\ \frac{\mu_{n}}{\sigma_{n}^{2}} &= \frac{n}{\sigma^{2}}\widehat{\mu_{n}} + \frac{ \mu_{0} }{ \sigma_{0}^{2} } \end{align}
where $\widehat{\mu_{n}}$ is the sample mean, $\frac{ 1 }{ n } \sum_{k=1}^{n} x_{k}$.
Thus:
$ p(\mu | \mathcal{D}) = \frac{ 1 }{ \sqrt{2} \sigma_{n} } \exp{\left[ -\frac{1}{2} \left( \frac{ \mu - \mu_{n} }{ \sigma_{n} } \right)^{2} \right]} $
Solving for $\mu_{n}$ and $\sigma_{n}^{2}$ yields:
\begin{align} \mu_{n} &= \left( \frac{ n\sigma_{0}^{2} }{ n\sigma_{0}^{2} + \sigma^{2} } \right) \widehat{\mu_{n}} + \frac{ \sigma^{2} }{ n\sigma_{0}^{2} + \sigma^{2} }\mu_{0} \\ \sigma_{n}^{2} &= \frac{ \sigma_{0}^{2} \sigma^{2} }{ n\sigma_{0}^{2} + \sigma^{2} } \end{align}
Okay! So… what? We’re trying to estimate the parameters of our sample distribution, and those parameters have their OWN distributions, e.g. $p(\mu | \mathcal{D})$.
The above equations show how the “parameters of the parameters” are related to the sample count $n$ and the mean and variance of the training samples.
After $n$ samples, $\mu_{n}$ is our best guess for $\mu$, and $\sigma_{n}^{2}$ is our uncertainty.
$ \sigma_{n}^{2} = \frac{ \sigma_{0}^{2}\sigma^{2} }{ n\sigma_{0}^{2} + \sigma^{2} } $
If this is our “uncertainty”, what happens as $n\rightarrow\infty$?
Our uncertainty goes towards zero! More samples means less uncertainty about our guess.
$ \mu_{n} = \left( \frac{ n\sigma_{0}^{2} }{ n\sigma_{0}^{2} + \sigma^{2} } \right) \widehat{\mu_{n}} + \frac{ \sigma^{2} }{ n\sigma_{0}^{2} + \sigma^{2} } \mu_{0} $
If this is our “best guess”, what happens as $n\rightarrow\infty$?
It is a linear combination of $\widehat{\mu_{n}}$ (the sample mean) and $\mu_{0}$ (our best prior guess for $\mu$).
As long as $\sigma_{0}^{2} \neq 0$, then $\mu_{n}$ approaches the sample mean as $n\rightarrow\infty$.
Our goal all along has been to obtain the probability of observing a sample value given the training set (associated with a particular class).
The message is our class-conditional probability is normally distributed:
$ p(x | \mathcal{D}) \sim N(\mu_{n}, \sigma^{2} + \sigma_{n}^{2}) $
In other words, we treat the “conditional mean” $\mu_{n}$ as if it was the mean of the class-conditional distribution, and the known variance is adjusted to account for the fact that there is some additional uncertainty about the value of the true mean.
If you didn’t follow any of the math (and aren’t planning on it), here are the main takeaway points:
Maximum Likelihood Estimation (MLE)
Bayesian Estimation (BAY)
As always, no single algorithm will have all the properties you want.
MLE makes some assumptions on the data and has some specific considerations:
Breaking down into direct comparisons:
These techniques for parameter estimation require a clear understanding of prior assumptions and a good use of existing data.
If your estimates seem to be performing worse than they should be, check both: Make sure the assumptions are valid (statistical analysis), and that your data are clean (plotting and quality control).
We will cover additional topics in the chapter (component analysis, dimensionality reduction, and Markov models) in some lectures after Spring Break.
We keep assuming we “know” our distributions, e.g. that they are Gaussian, Uniform, etc… but do we?
If the underlying distribution seems too complicated (high-dimensional, multi-modal), then we may need to throw away our precious Gaussians and look for something else.
There is a whole class of non-parametric techniques for density estimation, and a few methods – e.g. nearest-neighbor rules – that bypass underlying densities completely.


Robbie Williams has his say on Coleen Rooney-Rebekah Vardy feud
Robbie Williams has weighed in on Coleen Rooney’s feud with Rebekah Vardy.
Rooney, 33, made headlines around the world when she claimed Vardy’s Instagram account was the source of leaked stories about her private life.
Pop star Williams, 45, told the Telegraph Magazine: “There was a dramatic over-correction.
“Let’s say Rebekah Vardy was guilty, which we don’t know. How she will feel inside right now is 10 times worse than how Coleen Rooney would have felt when those things were happening to her.
“It’s a dangerous game to play…
“I understand Coleen Rooney’s need for revenge and I would probably have done the same thing. But we have to be very careful.
“It’s suicide-causing provoking those levels of shame.”
He warned that once a comment is out on social media it becomes beyond control and “a 16-headed monster”.
Williams said he has banned himself from knowing his own Twitter password, “because in anger I can be quite creative and my aggression would serve me badly”.
Vardy, 37, the wife of Leicester City striker Jamie, furiously denies any wrongdoing and said she has hired a cyber security expert to clear her name.
Rooney said she spent months trying to get to the bottom of who was leaking stories about her.
Vardy said arguing with Rooney would be like “arguing with a pigeon” as the fallout from their highly public dispute continued.
READ MORE
Coleen Rooney’s sleuthing skills praised as #WagathaChristie trends


NIGHT PRAYERSMake the sign of the cross,, Our Father…. Hail Mary….. Glory be…. I believe…. O my God, I thank you for all the benefits which I have ever received from you, especially this day, give me light to see what sins I have committed, and grant me grace to be truly sorry for them. Here examine your conscience for the faults you have committed during this day. O my God, because you are so good, I am very sorry that I have sinned against you and by the help of your grace I will not sin again. Into your hands I commend my spirit; Lord Jesus, receive my soul. Holy Mary, be a mother to me. May our Blessed Lady, St. Joseph, and all the Saints, pray for us to our Lord, that we may be preserved this night from sin and evil. O my good Angels, whom God has appointed to be my guardian, watch over me during this night. Angels and Saints of God, pray for me. . The Lord bless us and keep us from all evil and bring us to everlasting life. Amen. |


Half a year ago when I first boarding the on the Stem blockchain, I did not only dedicated most of my free time creating content on this amazing platform. But I also bought some Steem with around $2300 at the price of $1 by that time. Adding up all the SP and SBD I earned through the last 228 days by almost writing every day, my total SP is now at 3596.024.
This is of course just a negligible amount in the sea of Steem full of dolphins and whales. I used to think about it that way until I decided to look at how much I could earn if delegating all those SP away to some solid platforms like Smartsteem / @smartsteem.
Revenue Calculation
Without much troubles, Smartsteem provided a handy Revenue Calculation feature on their front page.
So I can roughly get a payout of 114 SBD every month with all my SP delegated to the platform. Math time again. Using current rate 1 SBD = $3.5
, 114*3.5 = $400
.
I could earn 400 USD every month while literally doing nothing at all now.
Let’s take it further
Back to my hometown in Melaka, fresh graduates starting pay is averaging about RM2000 per month. Not a really decent wages, but many of them just managed to survive anyhow.
While 400 USD equals to RM1558, I almost earn as much as those freshmen passively!
It is such a complicated mood while feeling sorry for the low pay that my fellow homie is making, easy revenue like this blew my mind away at the same time. Knowing the fact that we can actually cash out Steem or SDB make the whole thing even more real. I could really choose to resign my full-time job in Singapore and become half-retired in my hometown if I wanted.
Of course, I’m not retiring just for a $400 passive income
Nor I will delegate all SP away anytime soon as being totally powerless makes me feels like standing naked on the street. It is truly impressive to know that the initial investment of a $2300 and dedication in content creating could lead me to a pseudo-financial-freedom like this.
OK, I really have to stop thinking about how those whales could really live like a king, before the envious flame burn me down completely, haha!
Use my referral link to sign up Smartsteem if you found this article useful:
https://smartsteem.com?r=fr3eze
经过两百多天的努力和当初一笔价值 2300 美金的投资，我的有效 SP 终于来到 3596.024。这对各为大鱼面前来说简直不值一哂，我很多时候都是这样想的，那何不换个方法让自己开心点？
要是把这份 SP 完全租出去给类似 Smartsteem 的平台，会得到多少收入？答案是每个月约 114 SBD，换算成美金即是 $400。
这是什么概念？
拿我家乡马六甲来说，刚毕业出来的大学生们平均工资应该介于 2000 马币之间（很少，我懂）。而 $400 换成马币就是 RM1558，几乎都快赶上了他们的月收入。要知道，这可是什么都不用做的状态下的被动收入哦。
当然我是不打算完全把 SP 租出去的，那会让我感觉好像赤裸裸站在街上一般难受。但是换个角度一想，当初 $2300 的投资和大半年坚持写作的努力，居然隐约有可能让我回老乡进入半退休状态，还真的是不可思议。
如果你觉得这篇文章对你有帮助，请使用我的链接注册 Smartsteem：https://smartsteem.com?r=fr3eze
This page is synchronized from the post: ‘I’’m making a $400 passive income out of $2300 investment’


EventBufferEvent::connectHost (PECL event >= 1.2.6-beta) EventBufferEvent::connectHost — Connects to a hostname with optionally asyncronous DNS resolving 说明 public bool EventBufferEvent::connectHost ( EventDnsBase $dns_base , string $hostname , int $port [, int $family = EventUtil::AF_UNSPEC ] ) Resolves the DNS name hostname, looking for addresses of type family ( EventUtil::AF_* constants). If the name resolution fails, it invokes the event callback with an error event. If it succeeds, it launches a connection attempt just as EventBufferEvent::connect() would. dns_base is optional. May be NULL, or an object created with EventDnsBase::__construct() . For asyncronous hostname resolving pass a valid event dns base resource. Otherwise the hostname resolving will block. Note: EventDnsBase is available only if Event configured --with-event-extra ( event_extra library, libevent protocol-specific functionality support including HTTP, DNS, and RPC ). Note: EventBufferEvent::connectHost() requires libevent-2.0.3-alpha or greater. 参数 dns_base Object of EventDnsBase in case if DNS is to be resolved asyncronously. Otherwise NULL. hostname Hostname to connect to. Recognized formats are: www.example.com (hostname) <IP_ADDRESS> (ipv4address) ::1 (ipv6address) [::1] ([ipv6address]) port Port number family Address family. EventUtil::AF_UNSPEC , EventUtil::AF_INET , or EventUtil::AF_INET6 . See EventUtil constants . 返回值 Returns TRUE on success. Otherwise FALSE. 范例 Example #1 EventBufferEvent::connectHost() example input; //$bev->getInput(); //$pos = $input->search("TTP"); $pos = $bev->input->search("TTP"); while (($n = $bev->input->remove($buf, 1024)) > 0) { echo $buf; } } /* Event callback */ function eventcb($bev, $events, $base) { if ($events & EventBufferEvent::CONNECTED) { echo "Connected.\n"; } elseif ($events & (EventBufferEvent::ERROR | EventBufferEvent::EOF)) { if ($events & EventBufferEvent::ERROR) { echo "DNS error: ", $bev->getDnsErrorString(), PHP_EOL; } echo "Closing\n"; $base->exit(); exit("Done\n"); } } $base = new EventBase(); $dns_base = new EventDnsBase($base, TRUE); // We'll use async DNS resolving if (!$dns_base) { exit("Failed to init DNS Base\n"); } $bev = new EventBufferEvent($base, /* use internal socket */ NULL, EventBufferEvent::OPT_CLOSE_ON_FREE | EventBufferEvent::OPT_DEFER_CALLBACKS , "readcb", /* writecb */ NULL, "eventcb", $base ); if (!$bev) { exit("Failed creating bufferevent socket\n"); } //$bev->setCallbacks("readcb", /* writecb */ NULL, "eventcb", $base); $bev->enable(Event::READ | Event::WRITE); $output = $bev->output; //$bev->getOutput(); if (!$output->add( "GET {$argv[2]} HTTP/1.0\r\n". "Host: {$argv[1]}\r\n". "Connection: Close\r\n\r\n" )) { exit("Failed adding request to output buffer\n"); } if (!$bev->connectHost($dns_base, $argv[1], 80, EventUtil::AF_UNSPEC)) { exit("Can't connect to host {$argv[1]}\n"); } $base->dispatch(); ?> 以上例程的输出类似于： Connected. HTTP/1.0 301 Moved Permanently Location: http://www.google.co.uk/ Content-Type: text/html; charset=UTF-8 Date: Sat, 09 Mar 2013 12:21:19 GMT Expires: Mon, 08 Apr 2013 12:21:19 GMT Cache-Control: public, max-age=2592000 Server: gws Content-Length: 221 X-XSS-Protection: 1; mode=block X-Frame-Options: SAMEORIGIN


- Blog Posts
There are numerous issues in tech most of which are pretty well known to the general public. There's the issue of diversity in our industry, cyber-security, etc. However, the fact that very few people talk about net-neutrality is wild. Our internet is at risk of changing for the worse and many people don't even know it.
Net-Neutrality is defined as: "the principle that internet service providers should enable access to all content and applications regardless of the source, and without favoring or blocking particular products or websites." In essence, net-neutrality is the idea that all data on the internet should be treated the same.
If the concept still doesn't make sense to you this video can probably explain it better than I can.
Before I move on, it's also worth noting that in a crazy turn of events Tom Wheeler turned out to be a friend of the internet and passed new net-neutrality laws that would ensure the freedom of the internet. So he is in fact not a "Dingo" as John Oliver suggested in the link above.
Net Neutrality is a very important issue. We are all on the internet in one way or another every single day. We use the internet for leisure, business, etc. Needless to say, it's a HUGE part of our lives. Net-Neutrality as an issue has almost silently come and gone unbeknownst to everyone... and this even includes people in my generation (millenials) who are generally on top of most things tech.
Some may look at the new net-neutrality rules as putting unnecessary regulations on telecoms. Personally, I don't buy it. These new rules set the bar higher for ISP's and let's face it the bar was pretty low for them already. I can't recall anyone having rave reviews about their internet provider's customer service. I don't recall anyone having trouble from choosing their ISP from many options. Most people don't have an option or if they do it's between 2-4 providers max.
The underlying issue is what lobbying does to American politics (but that's an issue in itself). Allowing ISP's to create local monopolies has eliminated all incentive to deliver their absolute product to market. Not only that but services like Netflix, Hulu, etc pose a legitimate threat to traditional television so of course it's in their best interest to block Net Neutrality.
That to me is the final nail in the coffin. Not only are our internet speeds slow compared to the rest of the world but our ISP's are consistently coming in last place for customer-service. So remind me again why I should feel bad for cable television going out of style? I shouldn't, you shouldn't no one should. At this point it's best for us to be vigilant and fight for your internet rights. If your neighborhoods internet speeds are being throttled and you can prove it... you should go straight to the news about it. The laws have been made and it's up to all of us to hold our internet providers accountable. It's your internet, you're paying for it, don't settle.


In your external window you can either place your own designs, or simply use the built-in sdpi.css to style your content.
With a few lines of additional CSS you already get a nice info-window.
Clicking the button sends some information to your Property Inspector. Just open the remote-debugger for your Property Inspector and watch the console.


Sands of Iwo Jima (1949)
A Film Review by Mark O'Hara
John Wayne Sergeant John M. Stryker John Agar PFC Peter Conway Adele Mara Allison Bromley Forrest Tucker Corporal Al Thomas
SANDS OF IWO JIMA ranks as one of John Wayne's finest films, including his Westerns. It's story follows a tough platoon commander and his men through some of the bloodiest battles of the Pacific theater, including Tarawa and Iwo Jima.
Wayne's performance is a classic. I'm fairly sure it was my 10 year-old daughter's first exposure to Wayne, and I'm glad she saw him in a representative role. When you first see Wayne, he should be doing his best tough-guy act, like he does with Sergeant John M. Stryker. What a hard-as-nails name! (Isn't there a professional wrestler named Sergeant Striker?) To give a fictional character a middle initial is another nod to realism. Anyway, your first time seeing "The Duke" should be when he's unsentimental, doing a difficult job and shunning credit, all the while using his one-of-a-kind talk and walk.
Here his character was more challenging than many of his typical roles. Stryker is hard-nosed almost beyond tolerance, to the point that many of his men complain and begin the rumblings of mutiny. But when we find out that Stryker's wife has left him and taken their child, and that the boy does not write to his father, we see some of the motivation behind his behaviors. We understand better - though not completely - why he goes on drunken binges. Although the subplot concerning Stryker's personal life follows the formula of many war films, it certainly makes for good psychological study, and contributes a freshness to the film.
Two of Stryker's men take his treatment personally.  The old grudge surfaces until they are, once again, engaged in fisticuffs. Only this time a colonel spots the sergeant beating up on his own man, and we watch one Marine show his loyalty to another Marine, when Thomas claims Stryker was only instructing him in combat judo. In Thomas' confession to Stryker just after their make-up, we see the effects of a very sticky moral situation, a scene that would hold its own with any ethical dilemma I can think of in movies. Pfc. Peter Conway's hatred of Stryker is more complicated. He is a Harvard-educated businessman, an amateur soldier here to fight the war and then return home. When Conrad marries and later receives a letter from his wife, telling of their newborn baby, we see Stryker's jealousy. The only question at this point is how will Conway (played by John Agar) come to terms with his sergeant?
Real footage from Pacific campaigns is used convincingly in the film. When we see flame-throwing tanks igniting the battle-scarred hills of the islands, we form at least an idea of how devastating war can be. The smoothness of the film clips is pleasing too: the battlefields of the movie set match the real battlefields closely. And the difference in film quality is not much. Most remarkable is the planting of the flag atop Mount Suribachi. Director Alan Dwan uses an almost corny method of drawing our attention to the scene immortalized in the Iwo Jima Memorial - Marines struggling to raise the Stars and Stripes. But the effect caps the film nicely.
A recent trend in movie-making takes another look at standard genres and revises their supposed goals. Clint Eastwood's UNFORGIVEN showed us a Western whose hero was a gun-toting murderer. Steven Spielberg made perhaps the finest war movie ever in SAVING PRIVATE RYAN. My point is that RYAN has been advertised not as a war movie, but an "anti-war" movie. Taking a look at John Wayne's efforts in SANDS OF IWO JIMA, I cannot see how he or the film praises war. I cannot see, especially with the anti-climatic ending, how this narrative uses war for the mindless support of political causes.
I would recommend anyone watch this John Wayne film. And if you are watching with Wayne first-timers, withhold your comments about this grand old actor, and let them form their own opinions. The man is impressive enough to leave his mark on many generations.
The review above was posted to the
rec.arts.movies.reviews newsgroup (de.rec.film.kritiken for German reviews).
The Internet Movie Database accepts no responsibility for the contents of the
review and has no editorial control. Unless stated otherwise, the copyright
belongs to the author.
Please direct comments/criticisms of the review to relevant newsgroups.
Broken URLs inthe reviews are the responsibility of the author.
The formatting of the review is likely to differ from the original due
to ASCII to HTML conversion.
Related links: index of all rec.arts.movies.reviews reviews


(Color online) (A) side view and (B) top view of the optimized structure for the adsorption of H 2Pc on anatase (101) by using AE–LCAO (HSE) and centering the molecule on the Ti 5 c row. We represent Ti in blue, O in red, N in magenta, C in green, and H in orange.


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Activation
Product activation takes place when you enter your license code and click the “Verify” button (see Registering Hextrapolate).
The program connects once (and once only) to the wincent.com webserver to validate the license and the product is then activated permanently. If you later install the product on another machine, you will need to activate the product once more on that machine (see Transferring your license). The activation process takes only a few seconds.


ComputerSystem
Schema Definition
This schema defines a computer system and its respective properties. A computer system represents a machine (physical or virtual) and the local resources such as memory, cpu and other devices that can be accessed from that machine.Properties
| Property | Type | Nullable | Required | Permission | Description |
|---|---|---|---|---|---|
| @odata.context | String | read-only | The OData description of a payload. Format: uri | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| @odata.type | String | read-only | The type of a resource. | ||
| Actions { | Object | read-write | The available actions for this resource. | ||
| #ComputerSystem.Reset { | Object | read-write | This action is used to reset the system. | ||
| target | String | read-write | Link to invoke action Format: uri | ||
| title | String | read-write | Friendly action name | ||
| } | |||||
| Oem { } | Object | read-write | The available OEM specific actions for this resource. | ||
| } | |||||
| Boot { | Object | read-write | Information about the boot settings for this system. | ||
| BootSourceOverrideEnabled | String | Yes | read-write | Describes the state of the Boot Source Override feature. Possible Values:
| |
| BootSourceOverrideTarget | String | Yes | read-write | The current boot source to be used at next boot instead of the normal boot device, if BootSourceOverrideEnabled is true. Possible Values:
| |
| } | |||||
| Description | String | Yes | read-only | Provides a description of this resource and is used for commonality in the schema definitions. | |
| EthernetInterfaces { | Reference(EthernetInterfaceCollection) | read-only | A reference to the collection of Ethernet interfaces associated with this system. | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| Id | String | GET | read-only | Uniquely identifies the resource within the collection of like resources. | |
| IndicatorLED | String | Yes | read-write | The state of the indicator LED, used to identify the system. Possible Values:
| |
| Links { | Object | read-write | Contains references to other resources that are related to this resource. | ||
| Oem { } | Object | read-write | Oem extension object. | ||
| } | |||||
| Manufacturer | String | Yes | read-only | The manufacturer or OEM of this system. | |
| Model | String | Yes | read-only | The product name for this system, without the manufacturer name. | |
| Name | String | GET | read-only | The name of the resource or array element. | |
| Oem { | Object | read-write | This is the manufacturer/provider specific extension moniker used to divide the Oem object into sections. | ||
| EID_47597 { | Object | read-write | RECS|Box ComputerSystem extension | ||
| Baseboard { | Reference(Baseboard) | read-only | The Baseboard hosting this ComputerSystem | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| ComposedIn { | Reference(ComposedNode) | Yes | read-only | This is a link to the ComposedNode, this ComputerSystem is composed in. | |
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| FormFactor | String | read-only | The form factor of the computer system. Possible Values:
| ||
| HSLLInterface { | Reference(HSLLInterface) | Yes | read-only | This is a link to the HSLLInterface provided by this system | |
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| KvmStatus | String | read-write | The KVM status of the ComputerSystem. Attention: If set to KVM/iKVM, KVM is released from all other ComputerSystems. Possible Values:
| ||
| Power { | Reference(Power) | read-only | A reference to the power properties (power supplies, power policies, sensors) of this baseboard. | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| Thermal { | Reference(Thermal) | read-only | A reference to the thermal properties (fans, cooling, sensors) of this baseboard. | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| } | |||||
| } | |||||
| PowerState | String | Yes | read-only | This is the current power state of the system. Possible Values:
| |
| ProcessorSummary { | Object | read-write | This object describes the central processors of the system in general detail. | ||
| Count | Int64 | Yes | read-only | The number of physical processors in the system. Minimum: 0 | |
| Model | String | Yes | read-only | The processor model for the primary or majority of processors in this system. | |
| } | |||||
| Processors { | Reference(ProcessorCollection) | read-only | A reference to the collection of Processors associated with this system. | ||
| @odata.id | String | read-only | The unique identifier for a resource. Format: uri | ||
| } | |||||
| SKU | String | Yes | read-only | The manufacturer SKU for this system. | |
| SerialNumber | String | Yes | read-only | The serial number for this system. | |
| Status { | Object | read-write | This type describes the status and health of a resource and its children. | ||
| Health | String | Yes | read-only | This represents the health state of this resource in the absence of its dependent resources. Possible Values:
| |
| HealthRollup | String | Yes | read-only | This represents the overall health state from the view of this resource. Possible Values:
| |
| Oem { } | Object | read-write | Oem extension object. | ||
| State | String | Yes | read-only | This indicates the known state of the resource, such as if it is enabled. Possible Values:
| |
| } | |||||
| UUID | Guid | Yes | read-only | The universal unique identifier (UUID) for this system. Pattern: ([0-9a-fA-F]{8}-[0-9a-fA-F]{4}-[0-9a-fA-F]{4}-[0-9a-fA-F]{4}-[0-9a-fA-F]{12}) |
Actions
#ComputerSystem.Reset
Parameters
| Property | Type | Nullable | Required | Permission | Description |
|---|---|---|---|---|---|
| ResetType | String | Yes | read-write | The type of reset to be performed. Possible Values:
|
Request
POST /redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Actions/ComputerSystem.Reset
Content-Type: application/json
{
"ResetType": "On"
}
Content-Type: application/json
{
"ResetType": "On"
}
Response
HTTP/1.1 204
Location: Node
Location: Node
Operations
GET
Request
GET /redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1
Content-Type: application/json
Content-Type: application/json
Response
{
<EMAIL_ADDRESS>"uri",
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1",
<EMAIL_ADDRESS>"String",
"Actions": {
"#ComputerSystem.Reset": {
"target": "/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Actions/ComputerSystem.Reset",
<EMAIL_ADDRESS>["On", "ForceOff", "GracefulShutdown", "GracefulRestart", "ForceRestart", "ForceOn", "PushPowerButton"],
"title": "String"
}
"Oem": { }
}
"Boot": {
"BootSourceOverrideEnabled": "Disabled",
<EMAIL_ADDRESS>["Disabled", "Once", "Continuous"],
"BootSourceOverrideTarget": "None",
<EMAIL_ADDRESS>["None", "Pxe", "Cd", "Usb", "Hdd"]
}
"Description": "String",
"EthernetInterfaces": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/EthernetInterfaces"
},
"Id": "String",
"IndicatorLED": "Lit",
<EMAIL_ADDRESS>["Lit", "Blinking", "Off"],
"Links": {
"Oem": { }
}
"Manufacturer": "String",
"Model": "String",
"Name": "String",
"Oem": {
"EID_47597": {
"Baseboard": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1"
},
"ComposedIn": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/ComposedNodes/ComposedNode1"
},
"FormFactor": "Unknown",
"HSLLInterface": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/HSLLInterface"
},
"KvmStatus": "None",
<EMAIL_ADDRESS>["None", "KVM", "iKVM"],
"Power": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Power"
},
"Thermal": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Thermal"
}
}
}
"PowerState": "On",
"ProcessorSummary": {
"Count": 0,
"Model": "String"
}
"Processors": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Processors"
},
"SKU": "String",
"SerialNumber": "String",
"Status": {
"Health": "OK",
"HealthRollup": "OK",
"Oem": { },
"State": "Enabled"
}
"UUID": "00000000-0000-0000-0000-000000000000"
}
<EMAIL_ADDRESS>"uri",
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1",
<EMAIL_ADDRESS>"String",
"Actions": {
"#ComputerSystem.Reset": {
"target": "/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Actions/ComputerSystem.Reset",
<EMAIL_ADDRESS>["On", "ForceOff", "GracefulShutdown", "GracefulRestart", "ForceRestart", "ForceOn", "PushPowerButton"],
"title": "String"
}
"Oem": { }
}
"Boot": {
"BootSourceOverrideEnabled": "Disabled",
<EMAIL_ADDRESS>["Disabled", "Once", "Continuous"],
"BootSourceOverrideTarget": "None",
<EMAIL_ADDRESS>["None", "Pxe", "Cd", "Usb", "Hdd"]
}
"Description": "String",
"EthernetInterfaces": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/EthernetInterfaces"
},
"Id": "String",
"IndicatorLED": "Lit",
<EMAIL_ADDRESS>["Lit", "Blinking", "Off"],
"Links": {
"Oem": { }
}
"Manufacturer": "String",
"Model": "String",
"Name": "String",
"Oem": {
"EID_47597": {
"Baseboard": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1"
},
"ComposedIn": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/ComposedNodes/ComposedNode1"
},
"FormFactor": "Unknown",
"HSLLInterface": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/HSLLInterface"
},
"KvmStatus": "None",
<EMAIL_ADDRESS>["None", "KVM", "iKVM"],
"Power": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Power"
},
"Thermal": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Thermal"
}
}
}
"PowerState": "On",
"ProcessorSummary": {
"Count": 0,
"Model": "String"
}
"Processors": {
<EMAIL_ADDRESS>"/redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1/Processors"
},
"SKU": "String",
"SerialNumber": "String",
"Status": {
"Health": "OK",
"HealthRollup": "OK",
"Oem": { },
"State": "Enabled"
}
"UUID": "00000000-0000-0000-0000-000000000000"
}
POST
This operation is not supported on this resource and will result in the following response:
HTTP/1.1 405 Method Not Allowed
PATCH
Request
PATCH /redfish/v1/Chassis/Chassis1/Baseboards/Baseboard1/ComputerSystems/ComputerSystem1
Content-Type: application/json
{
"Boot": {
"BootSourceOverrideEnabled": "Disabled",
"BootSourceOverrideTarget": "None"
}
"IndicatorLED": "Lit",
"Oem": {
"EID_47597": {
"KvmStatus": "None"
}
}
}
Content-Type: application/json
{
"Boot": {
"BootSourceOverrideEnabled": "Disabled",
"BootSourceOverrideTarget": "None"
}
"IndicatorLED": "Lit",
"Oem": {
"EID_47597": {
"KvmStatus": "None"
}
}
}
Response
HTTP/1.1 204 No Content
DELETE
This operation is not supported on this resource and will result in the following response:
HTTP/1.1 405 Method Not Allowed


The Republic of the Walnut Islands is a sovereign island community for the priviledged few of our great nation, the United States of America, who desire a quiet life, free from regulation and relentless taxation. We are strong beleivers in the free market and the American Dream™.
Learn about the government, legal system, and borders of the Walnut Islands.
Learn about the climate and demographics of the Walnut Islands.
Learn about the language and culture of the Walnut Islands.
Learn about the economics and labour laws of the Walnut Islands.
Learn about Bay City, the central business district of the Walnut Islands.


Zhou, Ence and Hua, Song and Pi, Bingfeng and Sun, Jun and Nomura, Yashihide and Yamashita, Kazuhiro and Kurihara, Hidetoshi
NTMS, 2018
Security Assurance for Smart Contract
Exploratory
static analysis
source code => report
Any Ethereum
both


Message Rate Dialogue
This Dialogue box allows the user to rate the current open message.
First of all, the user has to decide, which interest is this message related to. In the "Inerest tree" list-box there are paths to 5 last visited interests in the interest tree. This is made for the user's convenience. If the needed interest hasn't been visited by the user recently, he/she has to find the path in the interest tree in the bottom-left part of the form. The Rating Edit box is locked until the interest is specified.
After the interest and its path are specified, the user has to grade the message from 1 to 10 and enter his/her grade in the Rating Edit box.
Button "Ok" returns the user to the Message_Body form upon adding the message rate to the database. Button "Cancel" returns the user to the Message_Body form without adding any info to the database.


Welcome!
Please view dashboard in full screen!
Explore the universities in Singapore
Explore the following:
- Find out the courses you are eligible for
- Find out the world and asia ranking of a university
- Find out the number of intakes and graduates
- Find out the overall employment rate and mean income of a degree
- Find out the admission grade of a degree


New Zealand’s Glaciers Turned Red (PHOTO)
Photographer Liz Carlson has shot coloured glaciers at Mount Aspiring National Park on New Zealand's South Island, Science Alert reported.
“Our glaciers don’t need any more battles as they are already truly endangered,” Carlson wrote on her blog. “It puts the impact of climate change into even more stark reality we can’t ignore.”
For now, it is unknown what caused the red colour of the glaciers. According to meteorologist Chris Brandolino, it could be soot or a carbon material from burnt wood. However, it could also just be red dust blown over from Australia, a phenomenon that's been observed before. Australian topsoil is rich in iron oxides, giving it its characteristic red hue.
In addition to the catastrophic fires on Australia's east coast, has also seen strong dust storms. Last month in Mildura, Victoria, a storm turned the sky red and the temperatures reached 40 degrees Celsius.
Whether the red cover of the pristine New Zealand mountains is dust, soot or anything else, Brandolino explained that several factors need to be combined to allow it to cross the Tasmanian Sea.
"You need to have smoke or dust in the air to begin with, and you have to have the right wind direction, and I think, having two happening at the same time, that's what's led [to this]”, he said.
"The fact that Australia had ... numerous fires out there, I think that's certainly been a part of it."
The photos also show how interconnected the ecosystems of our planet are. Pollution can even reach the most remote places.


| {{song.song}} | {{song.artist}} | {{song.album}} | ======={{song.song[:6]}}.. | | {{song.artist[:6]}}.. | | {{song.album[:6]}}.. | >>>>>>> parent of 6e5bbd7... update 8
|---|
| Sample | Sample | Sample |
|---|
| Song | Artist | Album |
|---|
| Sample | Sample | Sample |
|---|
| Song | Artist | Album |
|---|
| Sample | Sample | Sample |
|---|


Hi,
It was great meeting you at our booth, and we hope you enjoyed our Remote Inspection demo. Here is the picture we have taken with you!
To know more about our products and download a free trial:
TIBCO Software LimitedBusiness & Registered Address: Floor 23, 110 Bishopsgate, London, EC2N 4AY, United KingdomRegistered No.: 3875990 England & Wales| Phone +44<PHONE_NUMBER>Copyright and Privacy Policy


Splore Google API's
Map
Directions
Books
Enter a location of your choice (city, state, address)
SPLORE
Green lines
mark hipster bike-friendly lanes!


kotcity4
/
kotcity.data
/
Building
/
zots
zots
var
zots
:
List
<
Zot
>
List of
Zot
that this building suffers from. Populated by...
ZotPopulator


Home
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We'd love to hear from you.
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Open in maps


NFON REST API implementation for Node.js
The API addressed by this module is owned by NFON ©. I'm not affiliated.
npm install nfon
The first step you could do to test this module, is to get the current version of the API. This works without authentication.
import nfon from 'nfon';
nfon.api.version().then((res) => {
console.log(res);
});
If you want to make authenticated API calls, then you first need to configure your credentials from NFON.
import nfon from 'nfon';
nfon.config({
accessKeyId: 'xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx',
secretAccessKeyId: 'xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx'
})
const customerId = 'K1234'
nfon.api.GET('/api/customers/${customerId}/basic-data').then((res) => {
console.log(res);
});
There are 4 different request methods: POST, GET, PUT, DELETE
Here you can find more information about the API from NFON and also the current documentation with the different API requests: NFON Service Portal API
Set a direct dial call forward for a phone extension
This request sets the number<PHONE_NUMBER>789 as a call forward for the phone extension 100
import nfon from 'nfon';
const data: any = {
'links': [{
'rel': 'current',
'href': `/api/customers/K1234/forward-destinations/direct-dial/0048123456789`,
}],
};
nfon.api.PUT('/api/customers/K1234/targets/phone-extensions/100/call-forwards/DEFAULT', data).then((res) => {
console.log(res);
});
Get the direct dial call forward of a phone extension
This request gets the current call forward for the phone extension 100
import nfon from 'nfon';
nfon.api.GET('/api/customers/K1234/targets/phone-extensions/100/call-forwards/DEFAULT').then((res) => {
console.log(res);
});
This project is licensed under the MIT License - see the LICENSE.md file for details
Generated using TypeDoc


I'm currently a student here, studying Computer Science. It's a big change from my hometown, but so far I like it. :)
Mouse over for a special message!


@dimxy @Mihailo
{
"funcid": "s",
"txid": "b688049b059f6c4fd4819d5ac26bf04f66a994da5e40f4fdbc8db2cbc6714fe6",
"vout": 0,
"amount": "1000000",
"askamount": "1000000",
"origaddress": "RRbepNhmKmfFcWm3GuTqHxKY1VK1oTuLr6",
"tokenid": "cf6b8197cc60a7584850de8659542fbba6d64d41fdfe199b273728748ecdb59e",
"totalrequired": "1000000.00000000",
"price": "1.00000000"
},
{
"funcid": "S",
"txid": "0dde6ae2e1305bed2ba54076983d187464c95d0803731d904e1eda95a629cde7",
"vout": 0,
"amount": "990",
"askamount": "990",
"origaddress": "RRbepNhmKmfFcWm3GuTqHxKY1VK1oTuLr6",
"tokenid": "75ab46f823b2514f468a6132b52bfa463e1657ad9e83e65aaebbcdb822f8b56b"
}
It's possible to overfill a tokenfillask. I don't see it as much of a problem, but this is the output of tokenorders now. I could see this breaking dapps that make the assumption that "totalrequired": and "price" will always be there. Probably enough to just take note of it tbh


Neptune is the eighth and farthest known planet from the Sun in the Solar System. In the Solar System, it is the fourth-largest planet by diameter, the third-most-massive planet, and the densest giant planet. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times the mass of Earth and slightly larger than Neptune. Neptune orbits the Sun once every 164.8 years at an average distance of 30.1 astronomical units (4.50×109 km). It is named after the Roman god of the sea and has the astronomical symbol ♆, a stylised version of the god Neptune's trident.
Mean Distance from Sun(million km)
Period of Revolution
Period of Rotation at Equador
Eccentricity of Orbit
Equatorial diameter (km)
Mass
Density
Neptune
4,498.3
164.8 y
16 h
0.009
49,528
17.15
1.8
QUICK NEPTUNE FACTS
It takes Neptune 164.8 Earth years to orbit the Sun.
Neptune was discovered by Jean Joseph Le Verrier.
Neptune is the Roman God of the Sea. In Greek, Neptune is called Poseidon.
Neptune has the second largest gravity of any planet.
The largest Neptunian moon, Triton, was discovered just 17 days after Neptune itself was discovered.
Neptune has a storm similar the Great Red Spot on Jupiter.
Neptune also has a second storm called the Small Dark Spot.
Only one spacecraft, the Voyager 2, has flown past Neptune.
Like the other outer planets, Neptune possesses a ring system, though its rings are very faint.
Neptune has 14 known moons.
Neptune has an average surface temperature of -214°C – approximately -353°F.
DETAILED NEPTUNE FACTS
Neptune was not known to the ancients.It is not visible to the naked eye and was first observed in 1846. Its position was determined using mathematical predictions. It was named after the Roman god of the sea.
Neptune spins on its axis very rapidly.Its equatorial clouds take 18 hours to make one rotation. This is because Neptune is not solid body.
Neptune is the smallest of the ice giants.Despite being smaller than Uranus, Neptune has a greater mass. Below its heavy atmosphere Uranus is made of layers of hydrogen, helium, and methane gases. They enclose a layer of water, ammonia and methane ice. The inner core of the planet is made of rock.
The atmosphere of Neptune is made of hydrogen and helium, with some methane.The methane absorbs red light, which makes the planet appear a lovely blue. High, thin clouds drift in the upper atmosphere.
Neptune has a very active climate.Large storms whirl through its upper atmosphere, and high-speed winds track around the planet at up 600 meters per second. One of the largest storms ever seen was recorded in 1989. It was called the Great Dark Spot. It lasted about five years.
Neptune has a very thin collection of rings.They are likely made up of ice particles mixed with dust grains and possibly coated with a carbon-based substance.
Neptune has 14 moons.The most interesting moon is Triton, a frozen world that is spewing nitrogen ice and dust particles out from below its surface. It was likely captured by the gravitational pull of Neptune. It is probably the coldest world in the solar system.
Only one spacecraft has flown by Neptune.In 1989, the Voyager 2 spacecraft swept past the planet. It returned the first close-up images of the Neptune system. The NASA/ESA Hubble Space Telescope has also studied this planet, as have a number of ground-based telescopes.


---
layout: archive
title: Lean Prover Zulip Chat Archive
permalink: 113489newmembers/65068rflvsdectrivialforinequalityexample.html
---
The archive URL format has changed. Please find this page at:
https://leanprover-community.github.io/archive/stream/113489-new-members/topic/rfl.20vs.20dec_trivial.20for.20inequality.20example.html


Use this wizard to import one or more services into the Organization. First please choose a source of data to use for the import.
Tell us where your WADL file is located so we can load it up! We only need you to fill out *one* of the sections below.
Let us know which of the services we should import. It might also be a good idea to modify any of the services names we may have gotten wrong!
Choose which services you want to create from the list above.
Choose which plans will be available when Applications wish to create Service Contracts for the imported Service(s). Alternatively you can make the services "Public". Or else leave everything blank and then configure each imported Service later.


This module allows the administrator to manage the sensors and the user information.
System administrator can register new users for the Ocean Observation System.
Clicking the "Add an user" hyperlink will navigate the system administrator to the New User Register page.
You need to provide the following information:
* Choose one from "Administrator", "Scientist", "Data Curator". * Choose a Person from the database. See 3.3 Register New Person and 3.4 Update Personal Information for more information.
Click the "Add" button to register. Click the "cancel" button to cancel the registration.
System administrator can remove new users for the Ocean Observation System.
Clicking the "Remove an user" hyperlink will navigate the system administrator to the Manage Account page.
Click the "Remove" button to remove this user.
System administrators can enter the page which includes all information about users in the Ocean Observation System and manage all users information in this page.
Enter the user name and click the "GO" button. System administrator will be navagate to the "Manage Account" page and able to manage user's account.
You can only change:
Click the "Update" button to save the changes.
If administrators refuse to save the previous changes, click the "Cancel" button to give up the changes.
System administrators can register new person for the Ocean Observation System.
Clicking the "Add a Person" hyperlink will navigate system administrator to the New Person Register page.
You need to provide the following information:
* Note that the Email should be unique, which means that a person cannot have the same Email with any other person in the database.
Click the "Add" button to register. Click the "cancel" button to cancel the registration.
System administrator can update the personal information in the Ocean Observation System.
Select a person and click the "GO" button. System administrator will be able to manage personal information of this person.
You can change any of the following fields:
* Note that the Email should be unique, which means that a person cannot have the same Email with any other perseon in our database.
Click the "UPDATE" button to save the changes.
System administrators can remove person from the Ocean Observation System.
Clicking the "Remove a Person" hyperlink will navigate system administrator to the Manage Account page.
Administrators need to select a person to remove from Ocean Observation System
Click the "Remove" button to remove.
System administrator can register new sensor for the Ocean Observation System.
Clicking the "Add a sensor" hyperlink will navigate the system administrator to the New Sensor Register page.
You need to provide the following information:
* description is option, administrator can update this information later.
System administrators can remove sensor from the Ocean Observation System.
Clicking the "Remove a sensor" hyperlink will navigate system administrator to the Manage Sensor page.
Administrators need to select a sensor to remove from Ocean Observation System
Click the "Remove" button to remove.


Gloze interprets xml:id attributes as being of type xs:ID, regardless of whether or not a schema definition is available. The schema author must still design the schema so that the xml:id is a valid attribute. This may be done by adding it explicitly as a reference to the (imported) XML namespace schema; as xs:anyAttribute; or within xs:any.


"Video never lies, film does though." Blair Witch 2: Book of Shadows
By Ross Anthony
"There are a lot of nay-sayers who come in and ... say 'nay.'" This is the offhand, pseudo-documentary commentary that opens "Blair Witch" the sequel.  Well, not quite chilling horror; but eerie scary at least.
I'd wager to say this film is quite nearly similar to the first inentertainment value - but for different reasons. A hugely welcome 35mm camera on a hugely welcome tripod make the sequel several times more watchable than the first. No more dizziness, headaches or vomiting. That indelible image of grass and trees waving, scratching back and forth against the screen as our actor/cinematographers run amok, has been thankfully left as a first film phenomenon. Also, 98% of what you see on screen is not shot by the participants at all, but in normal film fashion with a real anonymous camera-operator. The editing, though a bit dodgy in the start, becomes quite normal.
Another welcome addition, the sequel has a bit more "plan" to it. Unfortunately, the plan never fully develops nor are the actors really scared. No matter what I read about that first film in the press notes ... it is still this reviewers opinion that those actors were truly riled and confused. And that captured on film is worth more than a very good plan. The sequel hosts no such real emotions. Though the actors are perhaps equally talented - the first batch didn't have to act as often.
Directing, acting and story being quite mediocre; it's the lack of climax that drops this film into the below average zone. Though the drama is actually pretty interesting, dare I say even mildly compelling, there's nothing meaningful that really ties the whole piece together at the end. Additionally, the motif of each player turning on the other was worked to death (pardon the statement) in the first film.
Unfortunately too, so passes the soul of the seed that is the "Blair Witch Project." Who cares about some child killing witch myth? It certainly wasn't this plot that brought the original film out of the woods and into cinema lore; it was the scheme! The truly riled actors, the ballsie hype, the simple idea.
This second film is more of the former and less of the later; making it more like any other ordinary scary movie of recent years. But in all fairness, for as much as it's departed from the first in guts, a sense of continuum maintains. And as with the first, there isn't much blood or slashing here - if that's the kind of thing you were hoping for, look elsewhere.
Here's the nutshell synopsis:
A year later in Burkittsville, the region the first "Project" claimed home to the murderous Blair Witch is now overrun with movie fans and tourists. One opportunist with a van and camera equipment, accompanied by four paying customers, sets out to camp at the foundation of that infamous unholy house of the first film. All in good fun, but as you might guess, they don't get much of a pleasant night's sleep, while more than just the cameras play tricks on them.
The Ending Contest: In 100 words or less write me a better ending to this film. (Shouldn't be too hard) I'll look over the entries and pick from them my personal favorite - sorry it's completely subjective. That winner will be printed on this site by Thanksgiving ... so send 'em in now. And let me look around, I can probably find a couple of free IMAX tickets around here somewhere to award the winner as well.
Blair Witch 2: Book of Shadows. Copyright © 2000. Rated R * Starring Kim Director, Jeffrey Donovan, Erica Leerhsen, Tristen Skyler, Stephen Barker Turner, Lanny Flaherty. * Directed by Joe Berlinger. * Written by Joe Berlinger and Dick Beebe. * Produced by Bill Carraro.
Grade..........................B-
-- Copyright © 2000. Ross Anthony, currently based in Los Angeles, has scripted and shot documentaries, music videos, and shorts in 35 countries across North America, Europe, Africa and Asia. For more reviews visit: http://www.RossAnthony.com
The review above was posted to the
rec.arts.movies.reviews newsgroup (de.rec.film.kritiken for German reviews).
The Internet Movie Database accepts no responsibility for the contents of the
review and has no editorial control. Unless stated otherwise, the copyright
belongs to the author.
Please direct comments/criticisms of the review to relevant newsgroups.
Broken URLs inthe reviews are the responsibility of the author.
The formatting of the review is likely to differ from the original due
to ASCII to HTML conversion.
Related links: index of all rec.arts.movies.reviews reviews


"Why" is Defined
It’s really surprising how often organizations fail to communicate “Why” work is important. At Labs, we want to make the “Why” explicit from day 1. Everything that happens on our Journey after this point should tie by to this “Why.”
Proven Ways to Define a Why
These are proven techniques to define your “Why”, but feel free to take your own approach if it
- Define a SMART Goal. We like the way Gojko Adzic articulates this in Impact Mapping1. We often use Impact Mapping as key facilitation technique during the Discovery stage.
- Define a Problem Statement2. 18F has a great guide, and the way they articulate the problem statement is perfectly acceptable to start an Impact Map
External References
- Page 5 is an example SMART Goal in Impact Mapping
- Defining a Problem Statement


Before Change
// the router lives. The rest of the hosts will be considered as remote workers.
try:
self._worker_hosts = [x.rsplit(":", 1)[0] if ":" in x else get_localhost_ip() for x in self._worker_procs]
self._max_workers = [int(x.rsplit(":", 1)[-1]) for x in self._worker_procs]
self._num_workers = [0 for x in self._worker_procs]
except:
raise RuntimeError(f"Incorrect format for option -j ({self._worker_procs}), which should be one or more [host:]nproc")
After Change
self._worker_hosts = []
self._max_workers = []
for worker_proc in self._worker_procs:
if ":" in worker_proc:
worker_host, max_workers = worker_proc.rsplit(":", 1)
if not max_workers.isdigit():
raise ValueError(f"Invalid worker specification {worker_proc}: number of process expected after ":"")
self._worker_hosts.append(worker_host)
self._max_workers.append(int(max_workers))
elif worker_proc.isdigit():
self._worker_hosts.append(get_localhost_ip())
self._max_workers.append(int(worker_proc))
else:
self._worker_hosts.append(worker_proc)
// here we assume that all nodes have the same number of cores so that we use
// the default value for master node for all computing nodes
self._max_workers.append(min(max(os.cpu_count() // 2, 2), 8))
self._num_workers = [0 for x in self._worker_procs]
except:
raise RuntimeError(f"Incorrect format for option -j ({self._worker_procs}), which should be one or more [host:]nproc")


|
Dendro
5.01
Dendro in Greek language means tree. The Dendro library is a large scale (262K cores on ORNL's Titan) distributed memory adaptive octree framework. The main goal of Dendro is to perform large scale multiphysics simulations efficeiently in mordern supercomputers. Dendro consists of efficient parallel data structures and algorithms to perform variational ( finite element) methods and finite difference mthods on 2:1 balanced arbitary adaptive octrees which enables the users to perform simulations raning from black holes (binary black hole mergers) to blood flow in human body, where applications ranging from relativity, astrophysics to biomedical engineering.
|
Public Member Functions | |
| MatRecord () | |
| The default constructor. | |
| unsigned int | getRowID () const |
| returns the row ID | |
| unsigned int | getColID () const |
| returns the col ID | |
| unsigned int | getRowDim () const |
| return the row dof | |
| unsigned int | getColDim () const |
| return the col dof | |
| void | setRowID (unsigned int rowID) |
| sets the rowID value | |
| void | setColID (unsigned int colID) |
| sets the rowID value | |
| void | setRowDim (unsigned int rowDim) |
| sets the rowDim value | |
| void | setColDim (unsigned int colDim) |
| sets the colDim value | |
| MatRecord (unsigned int rowID, unsigned int colID, unsigned int rowDim, unsigned int colDim, DendroScalar value) | |
| DendroScalar | getMatVal () const |
| returns the entry value | |
| void | setMatValue (DendroScalar value) |
| sets matrix value | |
| MatRecord (const MatRecord &other) | |
| The copy constructor. | |
| MatRecord & | operator= (MatRecord const &other) |
| The assignment operator. | |
| bool | operator== (MatRecord const &other) const |
| Overloaded == Operator. | |
| bool | operator!= (MatRecord const &other) const |
| Overloaded != Operator. | |
| bool | operator< (MatRecord const &other) const |
| Overloaded < Operator. | |
| bool | operator> (MatRecord const &other) const |
| Overloaded > Operator. | |
| bool | operator<= (MatRecord const &other) const |
| Overloaded <= Operator. | |
| bool | operator>= (MatRecord const &other) const |
| Overloaded >= Operator. | |
Friends | |
| std::ostream & | operator<< (std::ostream &os, MatRecord const &re) |
|
inline |
MatRecord constructor
| [in] | rowID | row id |
| [in] | colID | col id |
| [in] | rowDim | row dim |
| [in] | colDim | colDim |
| [in] | value | value of for the entry. |


Ocala events
-
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Hughes Homaid Chocolate Shop - Oshkosh, WI
11 Reviews of Hughes Homaid Chocolate Shop "Have been eating Hughes Chocolates since I was a . Oshkosh, WI 54902 . Oaks Candy Corner Whol & Ret .
http://www.yelp.com/biz/hughes-homaid-chocolate-shop-oshkoshLocations - Oaks Candy!
More than one hundred years and five generations later, chocolate lovers from all over still count on Oaks Candy for exceptional quality, value . Shop by category . Oaks, Inc. 1206 Oregon Street Oshkosh, WI 54902<EMAIL_ADDRESS>.
http://oakscandy.com/locations.html
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Oaks Candy Store - Oshkosh
Oshkosh, WI > Food Stores > Candy Stores > Oaks Candy Store .<PHONE_NUMBER> . 9 Waugoo Ave Oshkosh, WI 54901 . Hughes Home Maid Chocolate Shp .
http://www.insiderpages.com/b/3723637038/oaks-candy-store-oshkosh -
Hughes Home Maid Chocolate Shp - Oshkosh, WI 54902 - Intuit ...
920-231-7232. 1823 Doty St Oshkosh, WI 54902 . Features Candy Stores, Chocolate Manufacturers . Oaks candy (another Oshkosh staple) vs. Hughes.
http://business.intuit.com/directory/info-hughes-home-maid-chocolate-shp-oshkosh-wi-1 -
Source Wisconsin Candy Suppliers.
GIFT SHOP. Featured ads. El Club Mexicano Inc | Asheboro, .
http://www.macraesbluebook.com/search/product_company_list.cfm?prod_code=1164300®ion=Wisconsin-WI3 phase current monitor with ethernet
98 people's favorite
-
Oaks Candy Corner Whol & Ret - Oshkosh, WI
2 Reviews of Oaks Candy Corner Whol & Ret "I am writing yet another sweet tooth review. Thanks to . Oshkosh, WI 54902 . Hughes Homaid Chocolate Shop .
http://www.yelp.com/biz/oaks-candy-corner-whol-and-ret-oshkosh -
Oshkosh, Wisconsin - Wikipedia, the free encyclopedia
Oshkosh is a city in Winnebago County, Wisconsin, United States, located where . The fire had engulfed 70 stores, 40 factories, and 500 homes costing nearly . two well known chocolate companies located in Oshkosh, Hughes and Oaks.
http://en.wikipedia.org/wiki/Oshkosh,_Wisconsin


{% include 'includes/_head.html' %}
/api/v1/departments/mandatory-courses?key=YOUR_API_KEY
&department_code=CES
Note: This route also accepts methods DELETE and POST when accompanied by a JSON object of format {"department_code":"CES","course_code":"D101"}.
/api/v1/evalhalla/cities?key=YOUR_API_KEY
[&lang=fr
]
/api/v1/evalhalla/classifications?key=YOUR_API_KEY
/api/v1/evalhalla/departments?key=YOUR_API_KEY
[&lang=fr
]
/api/v1/offerings/counts-by-city?key=YOUR_API_KEY
&date_1=1960-01-01
[&date_2=1960-01-03
][&business_line=Digital Academy
][&clients_only=true
][&course_code=D101
][&exclude_cancelled=true
][&instructor_name=Paula
][&lang=fr
]
Note: Dates in standard YYYY-MM-DD ISO format.
Note: Parameters 'clients_only' and 'exclude_cancelled' are boolean.
/api/v1/offerings/offering-information?key=YOUR_API_KEY
&date_1=1960-01-01
[&date_2=1960-01-03
][&business_line=Digital Academy
][&clients_only=true
][&course_code=D101
][&exclude_cancelled=true
][&instructor_name=Paula
][&limit=20
][&offset=0
][&lang=fr
]
Note: Dates in standard YYYY-MM-DD ISO format.
Note: Parameters 'clients_only' and 'exclude_cancelled' are boolean.
/api/v1/registrations/course-codes?key=YOUR_API_KEY
[&lang=fr
]
/api/v1/registrations/department-codes?key=YOUR_API_KEY
[&lang=fr
]
/api/v1/registrations/training-locations?key=YOUR_API_KEY
&department_code=CES
/api/v1/tombstone/course_code
?key=YOUR_API_KEY
/api/v1/tombstone/course_code
/tombstone_value
?key=YOUR_API_KEY
Accepted values for 'tombstone_value':
Course Codes
To get the course codes that match the filters' criteria, use route:
/api/v1/comments/course-codes/
question_type
?key=YOUR_API_KEY
&department_code=CES
[&fiscal_year=1960-61
]Accepted values for 'question_type':
Note: Fiscal year in standard TBS hyphenated format.
Counts
To get the number of comments a course has received by stars, use route:
/api/v1/comments/counts/
question_type
?key=YOUR_API_KEY
&department_code=CES
[&course_code=D101
][&fiscal_year=1960-61
]Accepted values for 'question_type':
Note: Fiscal year in standard TBS hyphenated format.
Text
To get the raw text and attributes of comments a course has received, use route:
/api/v1/comments/text/
question_type
?key=YOUR_API_KEY
&department_code=CES
&limit=20
[&course_code=D101
][&fiscal_year=1960-61
][&stars=5
][&offset=0
][&lang=fr
]Accepted values for 'question_type':
Note: Accepted values for 'stars' are integers 1 through 5, inclusively.
Note: Fiscal year in standard TBS hyphenated format.


When you can able to view the Content of the site, Your Game will start. Actual game is basically you have to type the word shown above after the arrow(->) within 4 seconds.
The words are random and for each correct word your score will increase, What happens if you failed? that's okay!! just enter the current word shown, your game will restart!!
© Karthick


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Earrings
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Bengaluru, Karnataka - India


9 major APC lawmakers who have lost at appeal courts
What is happening in the Nigerian political space now is comparable to what happens in football: when a footballer scores, it's not yet time for full celebration, as it used to be, until the Video Assistant Referee (VAR), verifies, confirms or dismisses it.
Nigeria's 2019 elections have been concluded and winners have been announced but appeal courts are still hearing the petitions filed by candidates who were not satisfied with the outcome of some of the poll's results.
Significantly, the 'election VAR', that's the appeal courts, has been handing out some landmark rulings.
Some candidates declared by the Independent Electoral Commission (INEC) as winners have had their elections upturned by the court while some got theirs affirmed.
In this piece, Legit.ng highlights top lawmakers, Senators and House of Reps, on the platform of the All Progressives Congress (APC) whose elections have been upturned by the appeal courts.
Bayelsa/Kogi elections: Presidency urges PDP to accept defeat
1. Abdulmumin Jibrin, Kiru/Bebeji Constituency Federal Constituency, Kano
Abdulmumin Jibrin, a House of Reps member representing Kiru/Bebeji Constituency in Kano was sacked on Friday, November 1, by the appeal court and a fresh election was ordered within 90 days.
His election was challenged by Aliyu Datti Yako, the candidate of the Peoples Democratic Party (PDP).
2. Alhaji Bala Hassan, Sokoto North/Sokoto South Federal Constituency, Sokoto state
Alhaji Bala Hassan, representing Sokoto North/Sokoto South Federal Constituency, has also been sacked by the appeal court.
His election was challenged by the candidate of the PDP, Abubakar Abdullahi.
The Court of Appeal also ordered INEC to conduct rerun election within 90 days.
3. Mohammed Lawal, Ajaokuta Federal Constituency, Kogi state
Mohammed Lawal has equally been sacked as the lawmaker representing Ajaokuta Federal Constituency, Kogi state, in the House of Representatives.
Lawal's election was challenged by the PDP’s candidate, Aloysius Okino.
Goodluck Jonathan’s hometown calm hours to Bayelsa governorship election
The appeal court sitting in Abuja ordered INEC to conduct rerun elections in 21 polling units in the constituency.
4. Albert Akintoye, Okitipupa/Irele Federal Constituency, Ondo state
This is a bit different. The PDP candidate, Hon. Ikengboju Gboluga, was actually declared the winner by INEC.
But in a surprise ruling, the election tribunal sacked Gboluga and declared the APC candidate, Albert Akintoye, as the winner.
The tribunal sacked Gboluga for having dual citizenship, that is, for swearing to an oath of allegiance to the United Kingdom (UK).
However, the Court of Appeal later upturned the tribunal judgment and ordered INEC to issue Certificate of Return to Akintoye after withdrawing same from Gboluga.
The appeal court ruled that a Nigerian citizen by birth cannot be disqualified from contesting elections because he acquired additional citizenship of other countries to which he swore an oath of allegiance.
5. Senator Abubakar Tambuwal, Sokoto South
Inspiring story of a Nigerian chef who owns restaurants in US, cooking efo riro, egusi soup (photos)
Senator Abubakar Tambuwal's election was equally nullified by the appeal court. The court subsequently ordered the return of the PDP candidate, Ibrahim Danbaba, as the Senator representing the constituency.
6. Aliyu Shehu, Bodinga/Dange Shuni/Tureta Federal Constituency, Sokoto
The appeal court also nullified Shehu's election and ordered the return of PDP's Balarabe Kakale for the House of Representatives.
7. Alhassan Doguwa, Tudunwada/Doguwa federal constituency, Kano
The election of Alhassan Doguwa, the majority leader of the House of Representatives, has also been nullified by the appeal court.
Doguwa was sacked just as the court ordered a fresh election. His election was challenged by PDP's Yusha’u Salisu.
8. Emmanuel Uduaghan, former governor of Delta state and senator, Delta South
The Court of Appeal declared James Manager the winner of the Delta South Senatorial District election, voiding the judgement of the tribunal which had nullified Manager’s election and ordered a rerun.
9. Senator Adedayo Adeyeye, Ekiti South
In a judgement which has been described as shocking, the Court of Appeal sacked the spokesman of the Nigerian Senate, Adedayo Adeyeye, as the lawmaker representing Ekiti South senatorial district.
The court declared Senator Biodun Olujimi of the PDP winner of the election.
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Meanwhile, apparently determined to reconcile its aggrieved members which led to some needless defeats, the National Executive Committee (NEC) of the APC has adopted the recommendation of the national caucus that President Muhammadu Buhari should constitute a national reconciliation committee.
Legit.ng reports that the party in a thread of tweets on its Twitter page on Friday, November 22, said the standing committee would be empowered to address grievances of party members and resolve existing conflicts with a view to fostering unity and oneness within the party.
Premier League week three schedule: which teams will be battling it out?
NAIJ.com (naija.ng) -> Legit.ng We keep evolving to serve our readers better.
‘Nigerians have comprehensively rejected PDP’ | - on Legit TV


Arrays and Hashes
DBC Phase 0 Week 4
January 2, 2016
Arrays and hashes are both used to store collections in Ruby, but they are not always interchangeable.
Arrays store collections of objects, and can hold any type of object-strings, integers, floats, you can even have an array of arrays. But you cannot mix types within a single array.Arrays are ordered, meaning that they have some sort of order, even if they’re not completely sorted from smallest to largest. Each element in an array has an integer index which tells you where it is located in the array. The index is how you access an element’s value. It is important to remember that in Ruby (and most languages) the first index in an array is 0, and thus the last index is not equal to the number of elements in the array, but rather the number of elements – 1.
There are a few ways to declare an array in Ruby:
array_name = Array.new
Will create a new empty array without a specified size.
array_name = Array.new(5)
Will create a new empty array that has a size/length of 5 elements.
colors = ["red","green", "blue"]
Will create a new array with some of the elements already specified.
To access the value of an element use:
array_name[index]
For example,
colors[0]
Using the above array will return "red".
Hashes on the other hand, hold collections of key-value pairs. Instead of accessing each element by an integer index, you explicitly assign every element a key and a value. Keys are then used to access the value, and thus must be unique. This means you cannot use the same key for different values within the same hash, otherwise the program would not know which value you want when you reference a key. You can however, assign the same value to multiple keys. Both the key and the value can be any type of object, and you can use a mix of different objects within a single hash.
There are a few ways to declare a hash in Ruby:
hash_name = Hash.new
Will create a new empty hash, the default value for any new key will be nil, unless otherwise specified.
hash_name = Hash.new(value)
Will create a new empty hash, but now whatever is in parenthesis will be the default value.
colors = {"r" => "red", "g" => "green", "b" => "blue"}
Will create a new hash with those elements specified, where r, g, b, are keys and red, blue, green are their corresponding values.
To access the value of an element use:
hash_name[key]
For example,
colors["r"]
Using the above hash will return "red".


Auction clearances strengthen
by Unconventional EconomistAuction clearance rates lifted slightly over the weekend, with the preliminary rate nationally coming in at 74.1%, up from 72.0% last weekend:
Auction clearances were also way above the 42.0% recorded in the same weekend last year:
There are 1031 words left in this subscriber-only article.


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at
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10
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tall,
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placed
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very
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culms
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bluish-green
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where
covered
by
sheaths).
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to
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uppermost
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foliage;
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less
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most
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seen.
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new
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now.
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Level of confidence that token is an accurately-transcribed word
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As the only dedicated UI/UX Designer for the company, I collaborate closely with the CEO and Product Owners in identifying problems and coming up with solutions, Sprint planning, and etc. I work closely with the Development team and participate in the daily SCRUM meetings. I am primarily in charge of creating the user interface for the product and give inputs on the process and flow of the experience. Being in a startup company, I am used to working in a fast-paced environment and can adapt pretty well to sudden changes.
I create print and digital designs for promotional use of our product. I collaborated with the rest of the Creative Department and reported directly to the Chief Marketing Officer to come up with marketing campaigns for video ads, magazine ads, and sale promotions. I create design layouts such as graphics for social media postings, flyers, billboards, and other marketing colaterals. Also work as an in-company designer responsible for creating company business cards, business invitations, certificates, and other in-company needs.
I create graphic design for social media postings, billboards, standees and other marketing colaterals as requested by SM's Marketing Department.
I have been freelance designing for more than 5 years. I create logos, graphics design, web design, flyers, typography, and other design-related collaterals a client needs. I also develop simple websites for business landing pages, and others.
Google Developers Group is a non-profit organization that helps local I.T Communities. As a volunteer, I am responsible for Lorem ipsum dolor sit amet, which includes but not limited to planning, organizing, managing, and etc.
I create graphic design for social media postings, flyers, website mockups, typography and video animation as required by the various departments of the company, including Marketing and Administration.